#138569

order	512
nilpotency class	4
isoclinism	isoclinic class of #138568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9215
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

304 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,7,13,30,11,29,34,5,15,16,45,8,14,6,12)(3,9,20,37,21,47,33,46,22,39,31,43,18,38,19,36)(4,23,26,42,27,44,25,41,28,58,32,10,24,40,49,48)(17,50,53,62,55,63,52,61,51,64,56,35,54,57,59,60);(1,3,16,19,30,18,6,31,5,22,7,33,8,21,29,20)(2,9,34,37,14,47,13,46,15,39,12,43,11,38,45,36)(4,17,49,56,24,55,32,59,28,51,25,53,27,54,26,52)(10,35,40,50,48,61,23,57,42,62,44,64,41,60,58,63);(1,4)(2,10)(3,17)(5,28)(6,32)(7,25)(8,27)(9,35)(11,41)(12,44)(13,23)(14,48)(15,42)(16,49)(18,55)(19,56)(20,52)(21,54)(22,51)(24,30)(26,29)(31,59)(33,53)(34,40)(36,63)(37,50)(38,60)(39,62)(43,64)(45,58)(46,57)(47,61)

Polycyclic presentation:

2^9|a^2=f,b^2=df,d^2=ghi,e^2=h,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=eh,f^b=fhi,h^b=hi,d^c=dh,e^c=ehi,f^c=fhi,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45765);